Calculus

Calculus is an important branch of mathematics. The word stems from the ancient Greeks' use of pebbles arranged in patterns to study arithmetic and geometry. The Latin word for " pebble" is "calculus." Two complementary disciplines comprise calculus, both of which rely on the concept of a limit. The first is differential calculus, which is concerned with the instantaneous, as opposed to average, rate of change of a quantity. This can be illustrated by the slope of a function's graph at a particular point. The second is integral calculus, which studies the accumulation of infinitely small quantities, summing to areas under a curve, linear distance travelled, or volume displaced. These two processes act inversely to each other, as shown by the fundamental theorem of calculus.

Differential calculus typically provides a way to derive the acceleration and velocity of a body at a particular moment while integral calculus problems are used to compute areas and volumes, to find the amount of a liquid pumped by a pump with a set power input but varying conditions of pumping losses and pressure, or to find the amount of parking lot plowed by a snowplow of given power with varying rates of snowfall.

History

Historians generally regard integral calculus as going back no further than to the time of the ancient Greeks, circa 200 BC. Modern sources generally credit Hellenic mathematician Eudoxus with the method of exhaustion, which made it possible to compute the areas of regions and the volumes of solids. Archimedes developed this idea further, inventing heuristics which resemble integral calculus. After him, the development of calculus did not advance appreciably for over 500 years.

In India, the mathematician-astronomer Aryabhata in 499 used infinitesimals and expressed an astronomical problem in the form of a basic differential equation. Manjula, in the 10th century, elaborated on this differential equation in a commentary. This equation eventually led Bhaskara in the 12th century to develop independently a number of fundamental ideas in calculus, perhaps including an early form of the theorem now known as " Rolle's theorem". He was also the first to define the notion of the derivative as a limit. In the 14th century, Madhava, along with other mathematician-astronomers of the Kerala School, studied infinite series, power series, Taylor series, differentiation, integration, and the mean value theorem. Yuktibhasa, which some consider to be the first text on calculus, summarizes these results. These developments would not be duplicated in Europe until much later.

Sir Isaac Newton

Gottfried Wilhelm Leibniz

Calculus started making great strides in Europe towards the end of the early modern period and into the first years of the eighteenth century. This was a time of major innovation in Europe. Calculus provided a new method in mathematical physics to solve long-standing problems. Several mathematicians contributed to these breakthroughs, notably John Wallis and Isaac Barrow. James Gregory proved a special case of the second fundamental theorem of calculus in 1668. In Japan at around this time, Seki Kowa expanded further upon Eudoxus's method of exhaustion.

Leibniz and Newton pulled these ideas together into a coherent whole and they are usually credited with the probably independent and nearly simultaneous "invention" of calculus. Newton was the first to apply calculus to general physics and Leibniz developed much of the notation used in calculus today; he often spent days determining appropriate symbols for concepts. The fundamental insight that both Newton and Leibniz had was the fundamental theorem of calculus. Virtually all modern methods of symbolic integration follow from this theorem, and it proved indispensable in the development of modern mathematics and physics. For example, see Integration by parts and Integration by substitution.

When Newton and Leibniz first published their results, whether Leibniz's work was independent of Newton's was somewhat controversial. While Newton had derived his results years before Leibniz, Newton published only some time after Leibniz published in 1684. Later, Newton would claim that Leibniz got the idea from Newton's notes on the subject; however examination of the papers of Leibniz and Newton show they arrived at their results independently, with Leibniz starting first with integration and Newton with differentiation. This controversy between Leibniz and Newton divided English-speaking mathematicians from those in Europe for many years, which slowed the development of mathematical analysis. Today, both Newton and Leibniz are given credit for developing calculus independently. It is Leibniz, however, who is credited with giving the new discipline the name it is known by today: "calculus". Newton's name for it was "the science of fluxions". Some others who contributed important ideas are Descartes, Barrow, Fermat, Huygens, and Wallis.

Since the time of Leibniz and Newton, many mathematicians have contributed to the continuing development of calculus. In the 19th century, calculus was put on a much more rigorous footing by Cauchy, Riemann, Weierstrass, and others. It was also during this time period that the ideas of calculus were generalized to Euclidean space and the complex plane. Calculus continues to be further generalized, such as with the development of the Lebesgue integral in 1900.

Derivatives and Differentiation

The derivative measures the sensitivity of one variable to small changes in another variable. Consider the formula:

for an object moving at constant speed. The speed of a car, as measured by the speedometer, is the derivative of the car's distance traveled as a function of time. Calculus is a mathematical tool for dealing with this complex but natural and familiar situation.

The derivative of a curve defined by the function f(x) can be thought of as the slope, or angle, of the secant between two points on the curve at x and x+h, then letting the separation h between them shrink to zero.

Differential calculus can be used to determine the instantaneous speed at any given instant, while the formula "speed = distance divided by time" only gives the average speed, and cannot be applied to an instant in time because it then gives an undefined quotient zero divided by zero. Calculus avoids division by zero by using the concept of the limit which, roughly speaking, is a method of controlling an otherwise uncontrollable output, such as division by zero or multiplication by infinity. More formally, differential calculus defines the instantaneous rate of change (the derivative) of a mathematical function's value, with respect to changes of the variable. The derivative is defined as a limit of a difference quotient.

The derivative of a function, if it exists, gives information about its graph. It is useful for finding optimum solutions to problems, called maxima and minima of a function. It is proved mathematically that these optimum solutions exist either where the tangent of the graph is flat, so that the slope is zero; or where the graph has a sharp turn ( cusp) where the derivative does not exist; or at the endpoints of the graph. Another application of differential calculus is Newton's method, a powerful equation solving algorithm. Differential calculus has been applied to many questions that were first formulated in other areas, such as business or medicine.

The derivative lies at the heart of the physical sciences. Newton's second law of motion expressly uses the term "rate of change" which is the derivative: Therate of changeof momentum of a body is equal to the resultant force acting on the body and is in the same direction. Even the common expression of Newton's second law as: Force = Mass × Acceleration, involves differential calculus because acceleration is the derivative of velocity. (See Differential equation.) Maxwell's theory of electromagnetism and Einstein's theory of general relativity are also expressed in the language of differential calculus, as is the basic theory of electrical circuits and much of engineering. It is also applied to problems in biology, economics, and many other areas.

The derivative of a function y = f(x) with respect to x is usually expressed as either y ′ (read "y-prime"), f ' (x) (read "f-prime of x") or using Leibniz notation to write:

which is commonly shortened to:

Integrals and Integration

There are two types of integral in calculus, the indefinite and the definite. The indefinite integral is simply the antiderivative. That is, F is an antiderivative of f when f is a derivative of F. (This use of capital letters and lower case letters is common in calculus. The lower case letter represents the derivative of the capital letter.)

The definite integral evaluates the cumulative effect of many small changes in a quantity. The simplest instance is the formula

for calculating the distance a car moves during a period of time when it is traveling at constant speed. The distance moved is the cumulative effect of the small distances moved in each instant. Calculus is also able to deal with the natural situation in which the car moves with changing speed.

Integration can be thought of as measuring the area under a curve, defined by f(x), between two points (here a and b) by subdividing the area into ever-smaller slices and then adding them all up.

Integral calculus determines the exact distance traveled during an interval of time by creating a series of better and better approximations, called Riemann sums, that approach the exact distance as a limit. More formally, we say that the definite integral of a function on an interval is a limit of Riemann sum approximations.

Applications of integral calculus arise whenever the problem is to compute a number that is in principle (approximately) equal to the sum of a large number of small quantities. The classic geometric application is to area computations. In principle, the area of a region can be approximated by chopping it up into many pieces (typically rectangles, or, in polar coordinates, circular sectors), and then adding the areas of those pieces. The length of an arc, the area of a surface, and the volume of a solid can also be expressed as definite integrals. Probability, the basis for statistics, provides another important application of integral calculus.

The symbol of integration is ∫, a stretched s (which stands for "sum"). The precise meanings of expressions involving integrals can be found in the main article Integral. The definite integral, written as:

is read "the integral from a to b of f-of-x dx".

Foundations

There is more than one rigorous approach to the foundation of calculus. The usual one is via the concept of limits defined on the continuum of real numbers. An alternative is nonstandard analysis, in which the real number system is augmented with infinitesimal and infinite numbers. The tools of calculus include techniques associated with elementary algebra, and mathematical induction. The foundations of calculus are included in the field of real analysis, which contains all full definitions and proofs of the theorems of calculus as well as generalisations such as measure theory and distribution theory.

Fundamental theorem

The fundamental theorem of calculus states that differentiation and integration are, in a certain sense, inverse operations. More precisely, if one defines one function as the integral of another continuous function, then differentiating the newly defined function returns the function you started with. Furthermore, if you want to find the value of a definite integral, you usually do so by evaluating an antiderivative.

Here is the mathematical formulation of the Fundamental Theorem of Calculus: If a function f is continuous on the interval [a, b] and if F is a function whose derivative is f on the interval [a, b], then

Also, for every x in the interval [a, b],

This realization, made by both Newton and Leibniz, was key to the massive proliferation of analytic results after their work became known. The fundamental theorem provides an algebraic method of computing many definite integrals—without performing limit processes—by finding formulas for antiderivatives. It is also a prototype solution of a differential equation. Differential equations relate an unknown function to its derivatives, and are ubiquitous in the sciences.

Applications

The development and use of calculus has had wide reaching effects on all areas of modern living. It underlies nearly all of the sciences, especially physics. Virtually all modern developments such as building techniques, aviation, and other technologies make fundamental use of calculus. Many algebraic formulas now used for ballistics, heating and cooling, and other practical sciences were worked out through the use of calculus. In a handbook, an algebraic formula based on calculus methods may be applied without knowing its origins. The success of calculus has been extended over time to differential equations, vector calculus, calculus of variations, complex analysis, and differential topology. Calculus has been used on a broad field of subjects, even linguistics.

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